1. A study of blood pressure and measures of baseline stress recruited a random sample of l people. Baseline stress is recorded by detailed survey questions and a value from 1 [lowest] to 2|] {highest} is derived for the subject's selfreported stress. Subsequently, two diastolic blood pressure measurements {Y} are made on the subject on two diEerent visits. The data [datj comes in long format, with each blood pressure measurement per subject being on a diEerent row, with baseline stress {X} staying constant for both rows. The data analyst does the following command in R. lme~Ldata=datl a. Does this approach work for getting a reasonable esitmate of the change in El? per unit change in baseline stress? h. Assume the El? measurement are positively correlated within a subject. Do you think the resulting 95% condence interval from this procedure will have the right coverage, or be too big or too small? c. How could you do the analysis diEerently to insure that the standard error of the slope coe'icient estimate is not biased by ignoring the repeated measures on the same subject [more than one answer}? 2. A new related study is conducted where now, at the two visits, stress is also measured, so the data can be represented as {Xi = l, .. . ,m,j = 1,2}. {If interest in this study is to estimate how changes in stress are associated with changes in BP. 'Write out a linear regression model that addresses this question. Note, the researchers want to avoid treating the data as crosssectional and utilize the longitudinal nature to isolate diEerences due to baseline stress {Kn} and those due to changes in stress from the 1st to 2nd time. 3. Getting the estimate of interest from the coeictents. A study of asthma and air pollution recruited 21].] teenagers to report daily episodes of respiratory distress. In addition, an outside PM2.5 monitor was installed at their house and took daily measurements of PM2.5 as well. Finally, the participants reported the number of hours they went outside during the same day. The study lasted 3 months. They t the following logistic regression model: layit[P(Y.-j = IIXij1,Xs3-2}J= be + ~51ij1 + 523'va + bEXijl * Xijs where K1 is the respiratory distress indicator {= 1 if subject i had an episode on measurement j, and = D otherwise),X,-,jl is the log[PM2.5] measurements and ng-g is the number of hours subject spent outside. Assuming the above is the true model for the following. a. 1What is the odds ratio (in terms of the coefficients} of a change in X31 by 2 log units for students that spend 4 hours outside? h. 1What is the odds ratio for a change in X,,-1 by 2 log units among subjects that spend no time outside